OFFSET
0,3
COMMENTS
Based on characteristic polynomial of a square-within-a-square bonding graph.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,8,-36,-32,32,32).
FORMULA
From G. C. Greubel, Oct 05 2019: (Start)
a(2*n) = 4^(n+1)*(Fibonacci(2*n-2) + 1) - 2^n*(3*n-1).
a(2*n+1) = 2^(2*n+3)*(Fibonacci(2*n-1) - 1) + 2^(n+2)*n. (End)
MAPLE
seq(coeff(series(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 05 2019
MATHEMATICA
M = {{0, 1, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 1, 1, 0}, {1, 0, 1, 0, 0, 0, 1, 1}, {1, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0, 0, 0}}; f[x_] = Det[M - x*IdentityMatrix[8]]; Table[ SeriesCoefficient[ Series[x/(x^10*f[1/x]), {x, 0, 30}], n], {n, 0, 30}]
LinearRecurrence[{0, 12, 8, -36, -32, 32, 32}, {1, 0, 12, 8, 108, 160, 960}, 40] (* Harvey P. Dale, May 28 2017 *)
Table[If[EvenQ[n], 4^(n/2 +1)*(Fibonacci[n-2] +1) - 2^(n/2)*(3*n/2 -1), 2^(n+2)*(Fibonacci[n-2] -1) + 2^((n+1)/2)*(n-1)], {n, 0, 40}] (* G. C. Greubel, Oct 05 2019 *)
PROG
(PARI) my(x='x+O('x^40)); Vec(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2)) \\ G. C. Greubel, Oct 05 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-8*x^2-8*x^3)*(1-2*x^2)^2) )); // G. C. Greubel, Oct 05 2019
(Sage)
def A121961_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2)).list()
A121961_list(30) # G. C. Greubel, Oct 05 2019
(GAP) a:=[1, 0, 12, 8, 108, 160, 960];; for n in [8..40] do a[n]:=12*a[n-2] +8*a[n-3]-36*a[n-4]-32*a[n-5]+32*a[n-6]+32*a[n-7]; od; a; # G. C. Greubel, Oct 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Sep 02 2006
EXTENSIONS
Edited by N. J. A. Sloane, Feb 01 2007
STATUS
approved